Find the value of λ such that the vectors \(\vec{a}\) = 2\(\hat{i}\) + λ\(\hat{j}\) + \(\hat{k}\) and \(\vec{b}\) = \(\hat{i}\) + 2\(\hat{j}\) + 3\(\hat{k}\) are orthogonal
  • 0
  • 1
  • \(\frac{3}{2}\)
  • –\(\frac{5}{2}\)
The value of λ for which the vectors 3\(\hat{i}\) – 6\(\hat{j}\) + \(\hat{k}\) and 2\(\hat{i}\) – 4\(\hat{j}\) + λ\(\hat{k}\) are parallel is
  • \(\frac{2}{3}\)
  • \(\frac{3}{2}\)
  • \(\frac{5}{2}\)
  • –\(\frac{2}{5}\)
The vectors from origin to the points A and B are \(\vec{a}\) = 2\(\hat{i}\) – 3\(\hat{j}\) +2\(\hat{k}\) and \(\vec{b}\) = 2\(\hat{i}\) + 3\(\hat{j}\) + \(\hat{k}\) respectively, then the area of triangle OAB is
  • 340
  • \(\sqrt{25}\)
  • \(\sqrt{229}\)
  • \(\frac{1}{2}\) \(\sqrt{229}\)
For any vector \(\vec{a}\) the value of (\(\vec{a}\) × \(\vec{i}\))² + (\(\vec{a}\) × \(\hat{j}\))² + (\(\vec{a}\) × \(\hat{k}\))² is equal to
  • \(\vec{a}\)²
  • 3\(\vec{a}\)²
  • 4\(\vec{a}\)²
  • 2\(\vec{a}\)²
If |\(\vec{a}\)| = 10, |\(\vec{b}\)| = 2 and \(\vec{a}\).\(\vec{b}\) = 12, then the value of |\(\vec{a}\) × \(\vec{b}\)| is
  • 5
  • 10
  • 14
  • 16
The vectors λ\(\hat{i}\) + \(\hat{j}\) + 2\(\hat{k}\), \(\hat{i}\) + λ\(\hat{j}\) – \(\hat{k}\) and 2\(\hat{i}\) – \(\hat{j}\) + λ\(\hat{k}\) are coplanar if
  • λ = -2
  • λ = 0
  • λ = 1
  • λ = -1
If \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are unit vectors such that \(\vec{a}\) + \(\vec{b}\) + \(\vec{c}\) = \(\vec{0}\), then the value of \(\vec{a}\).\(\vec{b}\) + \(\vec{b}\).\(\vec{c}\) + \(\vec{c}\).\(\vec{a}\)
  • 1
  • 3
  • –\(\frac{3}{2}\)
  • None of these
Projection vector of \(\vec{a}\) on \(\vec{b}\) is
  • (\(\frac{\vec{a}.\vec{b}}{|\vec{b}|^2}\))\(\vec{b}\)
  • \(\frac{\vec{a}.\vec{b}}{|\vec{b}|}\)
  • \(\frac{\vec{a}.\vec{b}}{|\vec{a}|}\)
  • (\(\frac{\vec{a}.\vec{b}}{|\vec{a}|^2}\))\(\hat{b}\)
If \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are three vectors such that \(\vec{a}\) + \(\vec{b}\) + \(\vec{c}\) = 5 and |\(\vec{a}\)| = 2, |\(\vec{b}\)| = 3, |\(\vec{c}\)| = 5, then the value of \(\vec{a}\).\(\vec{b}\) +\(\vec{b}\).\(\vec{c}\) + \(\vec{c}\).\(\vec{a}\) is
  • 0
  • 1
  • -19
  • 38
If |\(\vec{a}\)| 4 and – 3 ≤ λ ≤ 2, then the range of |λ\(\vec{a}\)| is
  • [0, 8]
  • [-12, 8]
  • [0, 12]
  • [8, 12]
The number of vectors of unit length perpendicular to the vectors \(\vec{a}\) = 2\(\hat{i}\) + \(\hat{j}\) + 2\(\hat{k}\) and \(\vec{b}\) = \(\hat{j}\) + \(\hat{k}\) is
  • one
  • two
  • three
  • infinite
If (\(\frac{1}{2}\), \(\frac{1}{3}\), n) are the direction cosines of a line, then the value of n is
  • \(\frac{\sqrt{23}}{6}\)
  • \(\frac{23}{6}\)
  • \(\frac{2}{3}\)
  • –\(\frac{3}{2}\)
Find the magnitude of vector 3\(\hat{i}\) + 2\(\hat{j}\) + 12\(\hat{k}\)
  • \(\sqrt{157}\)
  • 4\(\sqrt{11}\)
  • \(\sqrt{213}\)
  • 9√3
Three points (2, -1, 3), (3, – 5, 1) and (-1, 11, 9) are
  • Non-collinear
  • Non-coplanar
  • Collinear
  • None of these
The vectors 3\(\hat{i}\) + 5\(\hat{j}\) + 2\(\hat{k}\), 2\(\hat{i}\) – 3\(\hat{j}\) – 5\(\hat{k}\) and 5\(\hat{i}\) + 2\(\hat{j}\) – 3\(\hat{k}\) form the sides of
  • Isosceles triangle
  • Right triangle
  • Scalene triangle
  • Equilateral triangle
The points with position vectors 60\(\hat{i}\) + 3\(\hat{j}\), 40\(\hat{i}\) – 8\(\hat{j}\) and a\(\hat{i}\) – 52\(\hat{j}\) are collinear if
  • a = -40
  • a = 40
  • a = 20
  • None of these
The ratio in which 2x + 3y + 5z = 1 divides the line joining the points (1, 0, -3) and (1, -5, 7) is
  • 5 : 3
  • 3 : 2
  • 2 : 1
  • 1 : 3
If O is origin and C is the mid point of A (2, -1) and B (-4, 3) then the value of \(\bar{OC}\) is
  • \(\hat{i}\) + \(\hat{j}\)
  • \(\hat{i}\) – \(\hat{j}\)
  • –\(\hat{i}\) + \(\hat{j}\)
  • –\(\hat{i}\) – \(\hat{j}\)
If ABCDEF is regular hexagon, then \(\vec{AD}\) + \(\vec{EB}\) + \(\vec{FC}\) is equal
  • 0
  • 2\(\vec{AB}\)
  • 3\(\vec{AB}\)
  • 4\(\vec{AB}\)
If \(\vec{a}\) = \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\), \(\vec{b}\) = 2\(\hat{i}\) – 4\(\hat{k}\), \(\vec{c}\) =\(\hat{i}\) + λ\(\hat{j}\) + 3\(\hat{j}\) are coplanar, then the value of λ is
  • \(\frac{5}{2}\)
  • \(\frac{3}{5}\)
  • \(\frac{7}{3}\)
  • –\(\frac{5}{3}\)
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